I'm a working mum of three, and also an employer, so I'm looking at it from both perspectives. It's bloody hard trying to make it work as a parent, but of course, it can also cause problems at work, so the best approach is to set a routine that doesn't change, so everyone - at home and at work - knows what's happening when so there's no last-minute surprises. Your manager knows where you are, so does your team, and at home, you can all settle into a routine.
Firstly, I'm sure you've mentioned it, but are you able to work from home at all, or is your role very much an office-based one? My starting point would be negotiating a day or two to work from home and offer it as a trial for them to see how effective it can be.
If WFH isn't doable as part of your usual working week, do you have flexibility with your hours? Are work open to condensed hours? So you can work longer days and shorter days? For example, on the days when DH is working from home and takes and collects the kids, can you do an early start, and a later finish to bank some hours? For example, if you worked 7.30 until 6, with a half hour lunch break, that's 10 hours. If you did this twice, you could then work shorter hours on the days when DH and you are both in the office?
On another of DH's work from home days, you could start early and finish early to pick them up. Say 7.30 start, take your lunch at 1.30 or 2, then you've done a full day. Or, you can make it shorter, and say you'll be back online between 4 and 5, in case anyone needs you (in case they need your role to be accessible).
Sound out your mum and dad to do one pick up per month and set the dates ahead of time, so everyone knows. Cook them a roast once a month to say thank you, but this is then one day, or maybe two, that you're getting extra help and relieving the pressure a little.
On one of the days your DH is in the office, can he start later and take the kids, and you go in early?
So, for example:
Monday - 7.30am-6pm - (DH at home) 10hrs
Tuesday - 7.30am-1.30pm, then 4-5pm (DH at home) 7hrs
Wednesday - 7.30 - 1.30pm, then 4-5pm (DH at home/office) 7hrs
Thursday - 9am-2.30pm, plus 4-5 (DH at home/in office) 6hrs
Friday - 9am-2.30pm, plus 4-5 (DH in office) 6hrs
When you pitch this, make it clear that while yes, this is benefitting you as a parent, you've been mindful to work it around particular needs of the business, so if Mondays are the busiest day, you've reflected this with your request. Are there particular things that you need to be in the office for, and other things you can really focus on in the quiet at home? In which case, evidence how you'll do these on a day where you're doing some of it at home - show them why this will be helpful for them, not just you.
Is there a HR dept you can talk ideas through with - they may suggest condensed hours, or show you what reducing hours in term time looks like, or other options?
As you rightly say, they want to keep you, soy need to make it easy for them to say yes. Let's face it, none of us are irreplaceable but there are some people genuinely worth their weight in gold and so it makes sense to look after them. I've made huge considerations for my team to reflect the demands they have outside of work, and as a result, they're hardworking and loyal, and know I'm always open to improving and changing things.
This is the hardest time to keep all plates spinning, I've been there and it's so difficult - I promise it gets easier. My youngest two are now teens and the shift is insane.
Good luck!